REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER

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1 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER DAMIR D. DZHAFAROV AND CARL MUMMERT Abstract. We study the reverse mathematcs of the prncple statng that, for every property of fnte character, every set has a maxmal subset satsfyng the property. In the context of set theory, ths varant of Tukey s lemma s equvalent to the axom of choce. We study ts behavor n the context of second-order arthmetc, where t apples to sets of natural numbers only, and gve a full characterzaton of ts strength n terms of the quantfer structure of the formula defnng the property. We then study the nteracton between propertes of fnte character and fntary closure operators, and the nteracton between these propertes and a class of nondetermnstc closure operators. 1. Introducton A formula ϕ wth one free set varable s of fnte character, and has the fnte character property, f ϕ( ) holds and, for every set A, ϕ(a) holds f and only f ϕ(f ) holds for every fnte F A. In ths paper, we restrct our attenton to formulas of second-order arthmetc, and consder several varants and restrctons of the prncple FCP (Defnton 2.1) whch asserts that for every formula of fnte character, every subset of N has a maxmal subset satsfyng that formula. Because the empty set satsfes any formula of fnte character, the soundness of ths prncple n second-order arthmetc can be verfed n ZFC by straghtforward applcaton of Zorn s lemma. Detaled defntons of second-order arthmetc and the subsystems studed n ths paper are gven by Smpson [4]. The prncple CE (Defnton 3.3) asserts that gven sets A B N, a formula ϕ of fnte character and a fntary closure operator D, such that A s a D-closed set satsfyng the formula, there s a set X whch s maxmal wth respect to the condtons that A X B, ϕ(x) holds, and X s D- closed. In the thrd secton, we gve a full characterzaton of the strength of fragments of CE n terms of the complexty of the formulas of fnte character to whch they apply. The authors are grateful to Dens Hrschfeldt, Antono Montalbán, and Robert Soare for valuable comments and suggestons, and to an anonymous referee who suggested an mprovement that strengthened Proposton 4.4. The frst author was partally supported by an NSF Graduate Research Fellowshp and an NSF Postdoctoral Fellowshp. 1

2 2 DAMIR D. DZHAFAROV AND CARL MUMMERT We can further generalze CE by replacng the fntary closure operator wth a more general knd of operator whch we name a nondetermnstc closure operator. The correspondng prncple, NCE (Defnton 4.2), s studed n the fnal secton, where a full characterzaton of ts strength s obtaned. We were led to study the reverse mathematcs of FCP by our separate work [1] on the prncple FIP whch states that every countable famly of subsets of N has a maxmal subfamly wth the fnte ntersecton property. All the prncples studed there are consequences of approprate restrctons of FCP. Smlarly, Propostons 3.7 and 4.4 below demonstrate how CE and NCE can be used to prove facts about countable algebrac objects n second-order arthmetc. In lght of these applcatons, we fnd t worthwle to have a complete understandng of the reverse mathematcs strengths of these prncples. Consderng ths paper together wth our work on FIP gves a new example of two prncples, FCP and FIP, whch are each equvalent to the axom of choce when formalzed n set theory, but whch have drastcally dfferent strengths when formalzed n second-order arthmetc. The axom scheme for FCP s equvalent to full comprehenson n second-order arthmetc, whle FIP s weaker than ACA 0 and ncomparable wth WKL Propertes of fnte character We begn wth the study of varous forms of the followng prncple. Defnton 2.1. The followng scheme s defned n RCA 0. (FCP) For each L 2 formula ϕ of fnte character, whch may have arbtrary set parameters, every set A has a -maxmal subset B such that ϕ(b) holds. FCP s analogous to the set-theoretc prncple M 7 n the catalog of Rubn and Rubn [3], whch s equvalent to the axom of choce [3, p. 34 and Theorem 4.3]. In order to better gauge the reverse mathematcal strength of FCP, we consder restrctons of the formulas to whch t apples. As wth other such ramfcatons, we wll prmarly be nterested n restrctons to classes n the arthmetcal and analytcal herarches. In partcular, for each {0, 1} and n 0, we make the followng defntons: Σ n-fcp s the restrcton of FCP to Σ n formulas; Π n-fcp s the restrcton of FCP to Π n formulas; n-fcp s the scheme whch says that for every Σ n formula ϕ(x) and every Π n formula ψ(x), f ϕ(x) s of fnte character and ( X)[ϕ(X) ψ(x)], then every set A has a -maxmal set B such that ϕ(b) holds. We also defne QF-FCP to be the restrcton of FCP to the class of quantferfree formulas wthout parameters. The followng proposton demonstrates two monotoncty propertes of formulas of fnte character.

3 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 3 Proposton 2.2. Let ϕ(x) be a formula of fnte character. The followng are provable n RCA 0 : (1) f A B and ϕ(b) holds then ϕ(a) holds; (2) f A 0 A 1 A 2 s a sequence of sets such that ϕ(a ) holds for each N, and N A exsts, then ϕ( N A ) holds. Proof. The proof of (1) s mmedate from the defntons. For (2), the key pont s to show that f F s a fnte subset of N A then there s some j N wth F A j. Ths follows from nducton on the Σ 0 1 formula ψ(n, F ) ( m)( < n)( F = A m ), n whch F s a set parameter. Our frst theorem n ths secton characterzes most of the above restrctons of FCP (see Corollary 2.5). We draw partcular attenton to part (2) of the theorem, where Σ 0 1 does not appear n the lst of classes of formulas. The reason behnd ths wll be made apparent by Theorem 2.6. Theorem 2.3. For {0, 1} and n 1, let Γ be any of Π n, Σ n, or n. (1) Γ-FCP s provable n Γ-CA 0 ; (2) If Γ s Π 0 n, Π 1 n, Σ 1 n, or 1 n, then Γ-FCP mples Γ-CA 0 over RCA 0. The proof of ths theorem wll make use of the followng techncal lemma, whch s needed only because there are no term-formng operatons for sets n the language L 2 of second-order arthmetc. For example, there s no term n L 2 that takes a set X and a number n and returns X D n where, as n the rest of ths paper, D n denotes the fnte set wth canoncal ndex n, or f n s not a canoncal ndex. The moral of the lemma s that such terms can be nterpreted nto L 2 n a natural way. The codng of fnte sets by ther canoncal ndces can be formalzed n RCA 0 n such a way that the predcate D n s defned by a formula ρ(, n) wth only bounded quantfers, and such that the set of canoncal ndces s also defnable by a bounded-quantfer formula [4, Theorem II.2.5]. Moreover, RCA 0 proves that every fnte set has a canoncal ndex. We use the notaton Y = D n to abbrevate the formula ( )[ Y ρ(, n)], along wth smlar notaton for subsets of fnte sets. Lemma 2.4. Let ϕ(x) be a formula wth one free set varable. There s a formula ϕ(x) wth one free number varable such that RCA 0 proves (2.4.1) ( A)( n)[a = D n = (ϕ(a) ϕ(n))]. Moreover, we may take ϕ to have the same complextes n the arthmetcal and analytc herarches as ϕ. Proof. Let ρ(, n) be the formula defnng the relaton D n, as dscussed above. We may assume ϕ s wrtten n prenex normal form. Form ϕ(n) by replacng each occurrence t X of ϕ, t a term, wth the formula ρ(t, n). Let ψ(x, Ȳ, m) be the quantfer-free matrx of ϕ, where Ȳ and m are sequences of varables that are quantfed n ϕ. Smlarly, let ψ(n, Ȳ, m) be the matrx of ϕ. Fx any model M of RCA 0 and fx n, A M such that

4 4 DAMIR D. DZHAFAROV AND CARL MUMMERT M = A = D n. proves that A straghtforward metanducton on the structure of ψ M = ( Ȳ )( m)[ψ(a, Ȳ, m) ψ(n, Ȳ, m)]. The key pont s that the atomc formulas n ψ(a, Ȳ, m) are the same as those n ψ(n, Ȳ, m), wth the excepton of formulas of the form t A, whch have been replaced wth the equvalent formulas of the form ρ(t, n). A second metanducton on the quantfer structure of ϕ shows that we may adjon quantfers to ψ and ψ untl we have obtaned ϕ and ϕ, whle mantanng logcal equvalence. Thus every model of RCA 0 satsfes (2.4.1). Because ρ has only bounded quantfers, the substtuton requred to pass from ϕ to ϕ does not change the complexty of the formula. We shall sometmes dentfy a fnte set wth ts canoncal ndex. Thus, f F s fnte and n s ts canoncal ndex, we may wrte ϕ(f ) for ϕ(n). Proof of Theorem 2.3. For (1), let ϕ(x) and A = {a : N} be an nstance of Γ-FCP. Defne g : 2 <N N {0, 1} by { 1 f ϕ({a j : τ(j) = 1} {a }) holds, g(τ, ) = 0 otherwse. where ϕ s as n the lemma. The functon g exsts by Γ comprehenson. By prmtve recurson, there exsts a functon h: N {0, 1} such that for all N, h() = 1 f and only f g(h, ) = 1. For each N, let B = {a j : j < h(j) = 1}. An nducton on ϕ shows that ϕ(b ) holds for every N. Let B = {a : h() = 1} = N B. Because Proposton 2.2 s provable n RCA 0 and hence n Γ-CA 0, t follows that ϕ(b) holds. By the same token, f ϕ(b {a k }) holds for some k then so must ϕ(b k {a k }), and therefore a k B k+1, whch means that a k B. Therefore B s -maxmal, and we have shown that Γ-CA 0 proves Γ-FCP. For (2), we assume Γ s one of Π 0 n, Π 1 n, or Σ 1 n; the proof for 1 n s smlar. We work n RCA 0 + Γ-FCP. Let ϕ(n) be a formula n Γ and let ψ(x) be the formula ( n)[n X = ϕ(n)]. It s easly seen that ψ s of fnte character, and t belongs to Γ because Γ s closed under unversal number quantfcaton. By Γ-FCP, N contans a -maxmal subset B such that ψ(b) holds. For any y, f y B then ϕ(y) holds. On the other hand, f ϕ(y) holds then so does ψ(b {y}), so y must belong to B by maxmalty. Therefore B = {y N : ϕ(y)}, and we have shown that Γ-FCP mples Γ-CA 0. The corollary below summarzes the theorem as t apples to the varous classes of formulas we are nterested n. Of specal note s part (5), whch says that FCP tself (that s, FCP for arbtrary L 2 -formulas) s as strong as any theorem of second-order arthmetc can be. Corollary 2.5. The followng are provable n RCA 0 :

5 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 5 (1) 0 1 -FCP, Σ0 0-FCP, and QF-FCP; (2) for each n 1, ACA 0 s equvalent to Π 0 n-fcp; (3) for each n 1, 1 n-ca 0 s equvalent to 1 n-fcp; (4) for each n 1, Π 1 n-ca 0 s equvalent to Π 1 n-fcp and to Σ 1 n-fcp; (5) Z 2 s equvalent to FCP. The case of FCP for Σ 0 1 formulas s anomalous. The proof of part (2) of Theorem 2.3 does not go through for Σ 0 1 because ths class s not closed under unversal quantfcaton. As the next theorem shows, ths lmtaton s qute sgnfcant. Intutvely, the proof uses the fact that a Σ 0 1 formula ϕ s contnuous n the sense that f ϕ(x) holds then there s an N such that ϕ(y ) holds for any Y wth X {0,..., N} = Y {0,..., N}. Theorem 2.6. Σ 0 1 -FCP s provable n RCA 0. Proof. Let ϕ(x) be a Σ 0 1 formula of fnte character. We clam that there exsts some c ϕ N such that for every set A, f A {0,..., c ϕ } = then ϕ(a) holds. To show ths, put ϕ(x) n normal form, so that ϕ(x) ( m)ρ(x[m]) where ρ s Σ 0 0. As ϕ( ) holds, there s some c = c ϕ such that ρ( [c]) holds. Now let A be any set such that A {0,..., c} =. Then ρ(a[c]) holds, so ϕ(a) holds. Ths proves the clam. Now fx any set A. By the clam, we know that ϕ(a {0,..., c ϕ }) holds. We may use bounded Σ 0 1 comprehenson [4, Theorem II.3.9] to form the set I of m such that D m {0,..., c ϕ } and ϕ(d m (A {0,..., c ϕ })) holds. We may then choose m I such that D m has maxmal cardnalty among the sets wth ndces n I. It follows mmedately that D m (A {0,..., c ϕ }) s a maxmal subset of A satsfyng ϕ. The above proof contans an mplct non-unformty n choosng a fnte set of maxmal cardnalty. The next proposton shows that ths nonunformty s essental, by showng that a sequental form of Σ 0 1-FCP s a strctly stronger prncple.

6 6 DAMIR D. DZHAFAROV AND CARL MUMMERT Proposton 2.7. The followng are equvalent over RCA 0 : (1) ACA 0 ; (2) for every famly A = A : N of sets, and every Σ 0 1 formula ϕ(x, x) wth one free set varable and one free number varable such that for all N, the formula ϕ(x, ) s of fnte character, there exsts a famly B = B : N of sets such that for all, B s a -maxmal subset of A satsfyng ϕ(x, ). Proof. The forward mplcaton follows by a straghtforward modfcaton of the proof of Theorem 2.3. For the reversal, let a one-to-one functon f : N N be gven. For each N, let A = {}, and let ϕ(x, x) be the formula ( y)[x X = f(y) = x]. Then, for each, ϕ(x, ) has the fnte character property, and for every set S that contans, ϕ(s, ) holds f and only f range(f). Thus, f B = B : N s the subfamly obtaned by applyng part (2) to the famly A = A : N and the formula ϕ(x, x), then It follows that the range of f exsts. range(f) B = {} B. Remark 2.8. Proposton 2.7 would not hold wth the class of boundedquantfer formulas of fnte character n place of the class of Σ 0 1 such formulas, because n that case part (2) s provable n RCA 0. Thus, n spte of the smlarty between the two classes suggested by the proof of Theorem 2.6, they do not concde. 3. Fntary closure operators We can strengthen FCP by mposng addtonal requrements on the maxmal set beng constructed. In partcular, we now consder requrng the maxmal set to satsfy a fntary closure property as well as a property of fnte character. Defnton 3.1. A fntary closure operator s a set of pars F, n n whch F s (the canoncal ndex for) a fnte (possbly empty) subset of N and n N. A set A N s closed under a fntary closure operator D, or D-closed, f for every F, n D, f F A then n A. Ths defnton of a closure operator s not the standard set-theoretc defnton presented by Rubn and Rubn [3, Defnton 6.3]. However, t s easy to see that for each operator of the one knd there s an operator of the other such that the same sets are closed under both. Our defnton has the advantage of beng readly formalzable n RCA 0. The followng prncple expresses the monotoncty of fntary closure operators. The proof follows drectly from defntons.

7 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 7 Proposton 3.2. It can be proved n RCA 0 that f D s a fntary closure operator and A 0 A 1 A 2 s a sequence of sets such that N A exsts and each A s D-closed, then N A s D-closed. The prncple n the next defnton s analogous to prncple AL 3 of Rubn and Rubn [3], whch s equvalent to the axom of choce n the context of set theory [3, p. 96, and Theorems 6.4 and 6.5]. Defnton 3.3. The followng scheme s defned n RCA 0. (CE) If D s a fntary closure operator, ϕ s an L 2 formula of fnte character, and A s any set, then every D-closed subset of A satsfyng ϕ s contaned n a maxmal such subset. In the termnology of Rubn and Rubn [3], ths s a prmed statement, meanng that t asserts the exstence not merely of a maxmal subset of a gven set, but the exstence of a maxmal extenson of any gven subset. Prmed versons of FCP and ts restrctons can be formed, and are equvalent to the unprmed versons over RCA 0. By contrast, CE has only a prmed form. Ths s because f A s a set, ϕ s a formula of fnte character, and D s a fntary closure operator, A need not have any D-closed subset of whch ϕ holds. For example, suppose ϕ holds only of, and D contans a par of the form, a for some a A. Ths leads to the observaton that the requrements n the CE scheme that the maxmal set must both be D-closed and satsfy a property of fnte character are, ntutvely, n opposton to each other. Satsfyng a fntary closure property s a postve requrement, n the sense that formng the closure of a set usually requres addng elements to the set. Satsfyng a property of fnte character can be seen as a negatve requrement n lght of part (1) of Proposton 2.2. We consder restrctons of CE as we dd restrctons of FCP above. By analogy, f Γ s a class of formulas, we use the notaton Γ-CE to denote the restrcton of CE to the formulas n Γ. We begn wth the followng analogue of part (1) of Theorem 2.3 from the prevous secton. Theorem 3.4. For {0, 1} and n 1, let Γ be Π n, Σ n, or 1 n. Then Γ-CE s provable n Γ-CA 0. Proof. Let ϕ be a formula of fnte character n Γ, whch may have parameters, and let D be a fntary closure operator. Let A be any set and let C be a D-closed subset of A such that ϕ(c) holds. For any X A, let cl D (X) denote the D-closure of X. That s, cl D (X) = N X, where X 0 = X and for each N, X +1 s the set of all n N such that ether n X or there s a fnte set F X such that F, n D. Because we take D to be a set, cl D (X) can be defned usng a Σ 0 1 formula wth parameter D. Defne a formula ψ(k, X) by ψ(k, X) ( n)[(d n cl D (X D k ) = ϕ(n)] cl D (X D k ) A,

8 8 DAMIR D. DZHAFAROV AND CARL MUMMERT where ϕ s as n Lemma 2.4. Note that ψ s arthmetcal f Γ s Π 0 n or Σ 0 n, and s n Γ otherwse. Defne a functon f : N {0, 1} nductvely such that f() = 1 f and only f ψ({j < : f(j) = 1} {}, C) holds. The characterzaton of the complexty of ψ ensures that ths f can be constructed usng Γ comprehenson, by frst formng the oracle {k : ψ(k, C)}. Now, for each N, let B = cl D (C {j < : f(j) = 1}), and let B = N B. The constructon of f ensures that ϕ(b ) mples ϕ(b +1 ) for all N, and we have assumed that ϕ holds of B 0 = cl D (C) = C. Therefore, an nstance of nducton shows that ϕ holds of B for all N, and thus also of B by Proposton 2.2. Ths also shows that B A. Smlarly, because each B s D-closed, the formalzed verson of Proposton 3.2 mples B s D-closed. Fnally, we check that B s maxmal. Suppose that H s a D-closed set such that B H A and ϕ(h) holds. Fxng H, because B B H and H s D-closed, we have cl D (B {}) H. Thus, ϕ(f ) holds for every fnte subset F of cl D (B {}), so by constructon f() = 1 and B +1 = cl D (B {}). Because B +1 B, we conclude that B. Thus B = H, as desred. It follows that for most standard syntactcal classes Γ, Γ-CE s equvalent to Γ-FCP. Indeed, for any class Γ we have that Γ-CE mples Γ-FCP, because any nstance of the latter can be regarded as an nstance of the former by addng an empty fntary closure operator. Conversely, f Γ s Π 0 n, Π 1 n, Σ 1 n, or 1 n, then Γ-FCP s equvalent to Γ-CA 0 by Theorem 2.3 (2), and hence equvalent to Γ-CE. Thus, n partcular, parts (2) (5) of Corollary 2.5 hold for CE n place of FCP, and the full scheme CE tself s equvalent to Z 2. The proof of the precedng theorem does not work for Γ = 0 1, because then Γ-CA 0 s just RCA 0, and we need at least ACA 0 to prove the exstence of the functon f defned there (the formula ψ(σ, X) beng arthmetcal at best). The next theorem shows that ths cannot be avoded, even for a class of consderably weaker formulas. Theorem 3.5. QF-CE mples ACA 0 over RCA 0. Proof. Assume a one-to-one functon f : N N s gven. Let ϕ(x) be the quantfer-free formula 0 / X, whch trvally has fnte character, and let p : N be an enumeraton of all prmes. Let D be the fntary closure operator consstng, for all, n N, of all pars of the form {p n+1 }, p n+2 ; {p n+2 }, p n+1 ; {p n+1 }, 0, f f(n) =. The set D exsts by 0 1 comprehenson relatve to f and our enumeraton of prmes.

9 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 9 Note that s a D-closed subset of N and ϕ( ) holds. Thus, we may apply CE for quantfer-free formulas to obtan a maxmal D-closed subset B of N such that ϕ(b) holds. By defnton of D, for every N, B ether contans every postve power of p or no postve power. Now f f(n) = for some n, then no postve power of p can be n B, because otherwse p n+1 would necessarly be n B and hence so would 0. On the other hand, f f(n) for all n then B {p n+1 : n N} s D-closed and satsfes ϕ, so by maxmalty p n+1 must belong to B for every n. It follows that range(f) f and only f p B, so the range of f exsts. The next corollary can be contrasted wth 2.5 part (1) and Theorem 2.6 to llustrate a dfference between CE from FCP n terms of some of ther weakest restrctons. Corollary 3.6. The followng are equvalent over RCA 0 : (1) ACA 0 ; (2) Σ 0 1 -CE; (3) Σ 0 0 -CE; (4) QF-CE. We conclude ths secton wth one addtonal llustraton of how formulas of fnte character can be used n conjuncton wth fntary closure operators. Recall the followng concepts from order theory: a countable jon-semlattce s a countable poset L, L wth a maxmal element 1 L and a jon operaton L : L L L such that for all a, b L, a L b s the least upper bound of a and b; an deal on a countable jon-semlattce L s a subset I of L that s downward closed under L and closed under L. The prncple n the followng proposton s the countable analogue of a varant of AL 1 n Rubn and Rubn [3]; compare wth Proposton 4.4 below. For more on the computablty theory of deals on lattces, see Turlngton [5]. Proposton 3.7. Over RCA 0, QF-CE mples that every proper deal on a countable jon-semlattce extends to a maxmal proper deal. Proof. Let L be a countable jon-semlattce. Let ϕ be the formula 1 X, and let D be the fntary closure operator consstng of all pars of the form {a, b}, c where a, b L and c = a b; {a}, b, where b L a. Because we defne a jon-semlattce to come wth both the order relaton and the jon operaton, the set D s 0 0 wth parameters, so RCA 0 proves D exsts. It s mmedate that a set X s closed under D f and only f X s an deal n L. We have not been able to prove a reversal correspondng to the prevous proposton.

10 10 DAMIR D. DZHAFAROV AND CARL MUMMERT Queston 3.8. What s the strength of the prncple assertng that every proper deal on a countable jon-semlattce extends to a maxmal proper deal? Ths queston s further motvated by work of Turlngton [5, Theorem ] on the smlar problem of constructng prme deals on computable lattces. However, because a maxmal deal on a countable lattce need not be a prme deal, Turlngton s results do not drectly resolve our queston. 4. Nondetermnstc fntary closure operators It appears that the underlyng reason that the restrcton of CE to arthmetcal formulas s provable n ACA 0 (and more generally, why Γ-CE s provable n Γ-CA 0 f Γ s as n Theorem 3.4) s that our defnton of fntary closure operator s very constranng. Intutvely, f D s such an operator and ϕ s an arthmetcal formula, and we seek to extend some D-closed subset B satsfyng ϕ to a maxmal such subset, we can focus largely on ensurng that ϕ holds. Achevng closure under D s relatvely straghtforward, because at each stage we only need to search through all fnte subsets F of our current extenson, and then adjon all n such that F, n D. Ths closure process becomes far less trval f we are gven a choce of whch elements to adjon. We now consder the case when each fnte subset F can be assocated wth a possbly nfnte set of numbers from whch we must choose at least one to adjon. Intutvely, ths change adds an aspect of dependent choce when we wsh to form the closure of a set. We wll show that ths weaker noton of closure operator leads to a strctly stronger analogue of CE. Defnton 4.1. A nondetermnstc fntary closure operator s a sequence of sets of the form F, S where F s (the canoncal ndex for) a fnte (possbly empty) subset of N and S s a nonempty subset of N. A set A N s closed under a nondetermnstc fntary closure operator N, or N-closed, f for each F, S n N, f F A then A S. Note that f D s a determnstc fntary closure operator, that s, a fntary closure operator n the stronger sense of the prevous secton, then for any set A there s a unque -mnmal D-closed set extendng A. Ths s not true for nondetermnstc fntary closure operators. For example, let N be the operator such that, N N and, for each N and each j >, {}, {j} N. Then any N-closed set extendng wll be of the form { N : k} for some k, and any set of ths form s N-closed. Thus there s no -mnmal N-closed set. In ths secton we study the followng nondetermnstc verson of CE. Defnton 4.2. The followng scheme s defned n RCA 0. (NCE) If N s a nondetermnstc closure operator, ϕ s an L 2 formula of fnte character, and A s any set, then every N-closed subset of A satsfyng ϕ s contaned n a maxmal such subset.

11 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 11 Because the unon of a chan of N-closed sets s agan N-closed, NCE can be proved n set theory usng Zorn s lemma. Restrctons of NCE to varous syntactcal classes of formulas are defned as for CE and FCP. Remark 4.3. We mght expect to be able to prove NCE from CE by sutably transformng a gven nondetermnstc fntary closure operator N nto a determnstc one. For nstance, we could go through the members of N one by one, and for each such member F, S add F, n to D for some n S (e.g., the least n). All D-closed sets would then ndeed be N-closed. The converse, however, would not necessarly be true, because a set could have F as a subset for some F, S N, yet t could contan a dfferent n S than the one chosen n defnng D. In partcular, a maxmal D-closed subset of a gven set mght not be maxmal among N-closed subsets. The results of ths secton demonstrate that t s mpossble, n general, to reduce nondetermnstc closure operators to determnstc ones n weak systems. Recall that an deal on a countable poset P, P s a subset I of P downward closed under P and such that for all p, q I there s an r I wth p P r and q P r. The next proposton s smlar to Proposton 3.7 above, whch dealt wth deals on countable jon-semlattces. In the proof of that proposton, we defned a determnstc fntary closure operator D n such a way that D-closed sets were closed under the jon operaton. For ths we reled on the fact that for every two elements n the semlattce there s a unque element that s ther jon. The reason we need nondetermnstc fntary closure operators below s that, for deals on countable posets, there are no longer unque elements wtnessng the relevant closure property. Proposton 4.4. Over RCA 0, QF-NCE mples that every deal on a countable poset can be extended to a maxmal deal. Proof. Let P, P be a countable poset; wthout loss of generalty we may assume P s nfnte. Form an extended poset P by adjonng a new element t to P and declarng q < P t for all q P. It follows mmedately that the deals on P correspond exactly to the deals of P that do not contan t, and each deal on P whch s maxmal among deals not contanng t corresponds to a maxmal deal on P. Fx an enumeraton {p : N} of P. We form a nondetermnstc closure operator N = N : N such that, for each N, f = 2 j, k and p j P p k then N = {p k }, {p j } ; f = 2 j, k, l + 1 and p j P p l and p k P p l then N = {p j, p k }, {p n : (p j P p n ) (p k P p n )} ; otherwse, N = {p }, {p }. Ths constructon gves a quantfer-free defnton of each N unformly n, so RCA 0 s able to construct N. Moreover, a subset of P s N-closed f and only f t s an deal.

12 12 DAMIR D. DZHAFAROV AND CARL MUMMERT Let ϕ(x) be the formula t X, whch s of fnte character. Fx an deal I P. Vewng I as a subset of P, we see that I s N-closed and ϕ(i) holds. Thus, by QF-NCE, there s a maxmal N-closed extenson J P satsfyng ϕ. Ths mmedately yelds a maxmal deal on P extendng I. Mummert [2, Theorem 2.4] showed that the proposton that every deal on a countable poset extends to a maxmal deal s equvalent to Π 1 1 -CA 0 over RCA 0, whch leads to the followng corollary. Ths contrasts sharply wth Theorem 3.4, whch showed that CE for arthmetcal formulas s provable n ACA 0. Corollary 4.5. QF-NCE mples Π 1 1 -CA 0 over RCA 0. We wll state the precse strength of QF-NCE n Corollary 4.7 below. We must frst prove the followng upper bound. The proof uses a technque nvolvng countable coded β-models, parallel to Lemma 2.4 of Mummert [2]. In ACA 0, a countable coded β-model s defned as a sequence M = M : N of subsets of N such that for every Σ 1 1 formula ϕ wth parameters from M, ϕ holds f and only f M = ϕ. Π 1 1 -CA 0 proves that every set s ncluded n some countable coded β-model. Complete nformaton on countable coded β-models s gven by Smpson [4, Secton VII.2]. Theorem 4.6. Σ 1 1 -NCE s provable n Π1 1 -CA 0. Proof. Let ϕ be a Σ 1 1 formula of fnte character (possbly wth parameters) and let N be a nondetermnstc closure operator. Let A be any set and let C be an N-closed subset of A such that ϕ(c) holds. Let M = M : N be a countable coded β-model contanng A, C, N, and any parameters of ϕ. Usng Π 1 1 comprehenson, we may form the set { : M = ϕ(m )}. Workng outsde M, we buld an ncreasng sequence B : N of N- closed extensons of C. Let B 0 = C. Gven, ask whether there s a j such that M j s an N-closed subset of A; B M j ; M j ; and ϕ(m j ) holds. If there s, choose the least such j and let B +1 = M j. Otherwse, let B +1 = B. Fnally, let B = N B. Because the nductve constructon only asks arthmetcal questons about M, t can be carred out n Π 1 1 -CA 0, and so Π 1 1 -CA 0 proves that B exsts. Clearly C B A. An arthmetcal nducton shows that for all N, ϕ(b ) holds and B s N-closed. Therefore, the formalzed verson of Proposton 2.2 shows that ϕ(b) holds, and the analogue of Proposton 3.2 for nondetermnstc fntary closure operators shows that B s N-closed.

13 REVERSE MATHEMATICS AND PROPERTIES OF FINITE CHARACTER 13 Now suppose that H s an N-closed set such that B H A and ϕ(h) holds. Fx H. Because ϕ s Σ 1 1, the property (4.6.1) ( X)[X s N-closed B X A X ϕ(x)] s expressble by a Σ 1 1 sentence wth parameters from M, and H wtnesses that t s true. Thus, because M s a β-model, ths sentence must be satsfed by M, whch means that some M j must also wtness t. The nductve constructon must therefore have selected such an M j to be B +1, whch means B +1 and hence B. It follows that B s maxmal. We can now characterze the strength of Σ 1 1-NCE and ts restrctons. Corollary 4.7. For each n 1, the followng are equvalent over RCA 0 : (1) Π 1 1 -CA 0; (2) Σ 1 1 -NCE; (3) Σ 0 n-nce; (4) QF-NCE. Proof. Theorem 4.6 shows that (1) mples (2), and t s obvous that (2) mples (3) and (3) mples (4). Corollary 4.5 shows that (4) mples (1). Our fnal results characterze the strength of NCE for formulas hgher n the analytcal herarchy. Theorem 4.8. For each n 1, (1) Σ 1 n-nce and Π 1 n-nce are provable n Π 1 n-ca 0 ; (2) 1 n-nce s provable n 1 n-ca 0. Proof. We prove part (1), the proof of part (2) beng smlar. Let ϕ(x) be a Σ 1 n formula of fnte character, respectvely a Π 1 n such formula. Let N be a nondetermnstc closure operator, let A be any set, and let C be an N-closed subset of A such that ϕ(c) holds. By Lemma 4.5, let ϕ be a Σ 1 n formula, respectvely a Π 1 n formula, such that ( X)( n)[x = D n = (ϕ(x) ϕ(n))]. We may use Π 1 n comprehenson to form the set W = {n : ϕ(n)}. Defne ψ(x) to be the arthmetcal formula ( n)[d n X = n W ]. We clam that for every set X, ψ(x) holds f and only f ϕ(x) holds. The defntons of W and ψ ensure that ψ(x) holds f and only f ϕ(d n ) holds for every fnte D n X, whch s true f and only f ϕ(x) holds because ϕ has fnte character. Ths establshes the clam. By the clam, ψ s a property of fnte character and ψ(c) holds. Usng Σ 1 1 -NCE, whch s provable n Π1 1 -CA 0 by Theorem 4.6 and thus s provable n Π 1 n-ca 0, there s a maxmal N-closed subset B of A extendng C wth property ψ. Agan by the clam, B s a maxmal N-closed subset of A extendng B wth property ϕ.

14 14 DAMIR D. DZHAFAROV AND CARL MUMMERT Corollary 4.9. The followng are provable n RCA 0 : (1) for each n 1, 1 n-ca 0 s equvalent to 1 n-nce; (2) for each n 1, Π 1 n-ca 0 s equvalent to Π 1 n-nce and to Σ 1 n-nce; (3) Z 2 s equvelent to NCE. Proof. The mplcatons from 1 n-ca 0, Π 1 n-ca 0, and Z 2 follow by Theorem 4.8. On the other hand, each restrcton of NCE trvally mples the correspondng restrcton of FCP, so the reversals follow by Corollary 2.5. Remark The characterzatons n ths secton shed lght on the role of the closure operator n the prncples CE and NCE. For n 1, we have shown that Σ 1 n-fcp, Σ 1 n-ce, and Σ 1 n-nce are all equvalent over RCA 0. However, QF-FCP s provable n RCA 0, QF-CE s equvalent to ACA 0 over RCA 0, and QF-NCE s equvalent to Π 1 1 -CA 0 over RCA 0. Thus the closure operators n the stronger prncples serve as a sort of replacement for arthmetcal quantfcaton n the case of CE, and for Σ 1 1 quantfcaton n the case of NCE. Ths allows these prncples to have greater strength than mght be suggested by the property of fnte character alone. At hgher levels of the analytcal herarchy, the prncples become equvalent because the complexty of the property of fnte character overtakes the complexty of the closure notons. References 1. Damr D. Dzhafarov and Carl Mummert, On the strength of the fnte ntersecton prncple, submtted, Carl Mummert, Reverse mathematcs of MF spaces, J. Math. Log. 6 (2006), no. 2, MR MR (2008d:03011) 3. Herman Rubn and Jean E. Rubn, Equvalents of the axom of choce. II, Studes n Logc and the Foundatons of Mathematcs, vol. 116, North-Holland Publshng Co., Amsterdam, MR MR (87c:04004) 4. Stephen G. Smpson, Subsystems of second order arthmetc, second ed., Perspectves n Logc, Cambrdge Unversty Press, Cambrdge, MR MR (2010e:03073) 5. Amy Turlngton, Computablty of Heytng algebras and dstrbutve lattces, Ph.D. dssertaton, Unversty of Connectcut, Department of Mathematcs, Unversty of Notre Dame, Department of Mathematcs, Unversty of Notre Dame, 255 Hurley Hall, Notre Dame, Indana USA E-mal address: ddzhafar@nd.edu Department of Mathematcs, Marshall Unversty, 1 John Marshall Drve, Huntngton, West Vrgna USA E-mal address: mummertc@marshall.edu